Geometry and Physics
This is a selective course for Bachelor's students in the programs Mathematics, Technical Mathematics, Astronomy, and Physics & Mathematics and for MSc students in the programs Physics and Applied Physics.
Gauss and Riemann, among others, studied the properties of manifolds: curved, non-Euclidean spaces. But it was Einstein's theory of General Relativity that familiarized physicists with such spaces. Differential forms are a powerful tool for computing on manifolds, while at the same time they offer a beautiful geometric abstraction. After introducing the concept of a submanifold of a Euclidean space we discuss vector fields and differential forms in Euclidean space. Finally, we generalize these concepts to manifolds. We show how the generalized Stokes theorem expressed in the language of differential forms unifies the, well-known from vector calculus, theorems of Gauss and Stokes. Finally, we discuss applications of differential forms in electromagnetism and dynamical systems.
- Meetkunde en fysica, met differentiaalvormen en integraalstellingen, H. W. Broer, Epsilon Uitgaven, 1999.